text: For general application of the Rietveld method, irrespective of the software used, the observed Bragg peaks in a powder diffraction pattern are best described by the so-called peak shape function (PSF). The PSF is a convolution of three functions: the instrumental broadening , wavelength dispersion , and the specimen function , with the addition of a background function, . It is represented as follows:
  where  denotes convolution, which is defined for two functions  and  as an integral:
  The instrumental function depends on the location and geometry of the source, monochromator, and sample. Wavelength function accounts for the distribution of the wavelengths in the source, and varies with the nature of the source and monochromatizing technique. The specimen function depends on several things. First is dynamic scattering, and secondly the physical properties of the sample such as crystallite size, and microstrain.
  A short aside: unlike the other contributions, those from the specimen function can be interesting in materials characterization. As such, average crystallite size, , and microstrain, , effects on Bragg peak broadening,  (in radians), can be described as follows, where  is a constant:
  : and .
  Returning to the peak shape function, the goal is to correctly model the Bragg peaks which exist in the observed powder diffraction data. In the most general form, the intensity, , of the  point (, where  is the number of measured points) is the sum of the contributions  from the  overlapped Bragg peaks (), and the background, , and can be described as follows:
  where  is the intensity of the  Bragg peak, and . Since  is a multiplier, it is possible to analyze the behaviour of different normalized peak functions  independently of peak intensity, under the condition that the integral over infinity of the PSF is unity. There are various functions that can be chosen to do this with varying degrees of complexity. The most basic functions used in this way to represent Bragg reflections are the Gauss, and Lorentzian functions. Most commonly though, is the pseudo-Voigt function, a weighted sum of the former two (the full Voigt profile is a convolution of the two, but is computationally more demanding). The pseudo-Voigt profile is the most common and is the basis for most other PSF's. The pseudo-Voigt function can be represented as:
  where
  and
  are the Gaussian and Lorentzian contributions, respectively.
  Thus,
  where:
  *  and  are the full widths at half maximum (FWHM)
  * is essentially the Bragg angle of the point in the powder pattern with its origin in the position of the peak divided by the peak's FWHM.
  *  ,  and  and  are normalization factors such that  and  respectively.
  * , known as the Caglioti formula, is the FWHM as a function of  for Gauss, and pseudo-Voigt profiles. , , and  are free parameters.
  * is the FWHM vs.  for the Lorentz function.  and  are free variables
  * , where  is the pseudo-Voigt mixing parameter, and  are free variables.
  The pseudo-Voigt function, like the Gaussian and Lorentz functions, is a centrosymmetric function, and as such does not model asymmetry. This can be problematic for non-ideal powder XRD data, such as those collected at synchrotron radiation sources, which generally exhibit asymmetry due to the use of multiple focusing optics.
  The Finger–Cox–Jephcoat function is similar to the pseudo-Voigt, but has better handling of asymmetry, which is treated in terms of axial divergence. The function is a convolution of pseudo-Voigt with the intersection of the diffraction cone and a finite receiving slit length using two geometrical parameters, , and , where  and  are the sample and the detector slit dimensions in the direction parallel to the goniometer axis, and  is the goniometer radius.